Summary
The Lewis-Milne (LM) equation has been widely applied for design of border irrigation systems. This equation is based on the concept of mass conservation while the momentum balance is replaced by the assumption of a constant surface water depth. Although this constant water depth depends on the inflow rate, slope and roughness of the infiltrating surface, no explicit relation has been derived for its estimation. Assuming negligible border slope, the present study theoretically treats the constant depth in the LM equation by utilizing the simple dam-break wave solution along with boundary layer theory. The wave front is analyzed separately from the rest of the advancing water by considering both friction and infiltration effects on the momentum balance. The resulting equations in their general form are too complicated for closed-form solutions. Solutions are therefore given for specialized cases and the mean depth of flow is presented as a function of the initial water depth at the inlet, the surface roughness and the rate of infiltration. The solution is calibrated and tested using experimental data.
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Abbreviations
- a (t) :
-
advance length
- c :
-
mean depth in LM equation
- c f :
-
friction factor
- c h :
-
Chezy's friction coefficient
- g :
-
acceleration due to gravity
- h(x, t) :
-
water depth
- h 0 :
-
water depth at the upstream end
- i(τ):
-
rate of infiltration
- f(x, t):
-
discharge
- q0 :
-
constant inflow discharge
- S f :
-
energy loss gradient or frictional slope
- S0 :
-
bed slope
- t :
-
time
- u(x, t) :
-
mean velocity along the water depth
- x :
-
distance
- Y(τ):
-
cumulative infiltration
- η(t) :
-
distance separating two flow regions
- τ:
-
infiltration opportunity time
References
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Singh, V.P., Scarlatos, P.D. & Prasad, S.N. An improved Lewis-Milne equation for the advance phase of border irrigation. Irrig Sci 11, 1–6 (1990). https://doi.org/10.1007/BF00189988
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DOI: https://doi.org/10.1007/BF00189988